What does the notation $\nabla_\theta \mathcal{L}$ mean?
This is standard backpropagation. The gradient term you see is in fact a vector of partial derivatives where each element is the partial derivative of the log-likelihood with respect to each element of the parameter vector $\theta$. Therefore, it has the same dimensionality as $\theta$. Each element of the parameter vector is then updated with the respective term in the vector of partial derivatives, which are generally not the same.